Micro-electromechanical (MEMs) resonators that are operated in a lateral bulk extension mode may have several critical parameters that can influence resonator operating frequency. Some of these critical parameters can be highlighted by modeling performance of a resonator using a simplified bulk acoustic wave equation: f=v/(2L), where f is a resonant frequency, v is an acoustic velocity of the resonator material and L is the lateral dimension of a resonator body along an axis of vibration. For a bulk acoustic resonator containing a resonator body with a composite stack of layers thereon, the acoustic velocity is a function of the Young's modulus, density and thickness of the resonator body and each of the stack of layers.
Accordingly, because the thicknesses of all of the layers may vary during deposition processes, variations in resonant frequency may be present between otherwise equivalent devices formed across a wafer(s). For example, variations in thicknesses of 1-2% across a wafer may cause significant deviations in frequency on the order of several thousands of parts-per-million (ppm).
The same is true for process-induced variations in the lateral dimensions of the resonator body, which may be caused by photolithographic and etching variations across a substrate (e.g., wafer) and batch processing of multiple substrates. These variations in lateral dimension can come from variations in the photolithographic patterning of the resonator body and from variations in sidewall angle during etching processes that separate the resonator body from a surrounding substrate. Unfortunately, these dimensional variations may cause a frequency drift on the order of several thousand ppm for a resonator operating in the megahertz resonant frequency range.
Similarly, a thin-film bulk acoustic resonator may be subject to temperature-induced frequency drift in resonant frequency, which is caused by materials (e.g., Si, AlN, Mo) within the resonator that have a negative temperature coefficient of frequency (TCF), resulting from a reduction in respective elastic constants as temperature increases. For example, an acoustic resonator having an even relatively small TCF value per degree of temperature variation, may have an unacceptably large resonant frequency variation across a range of normal operating temperatures.